Journal of Mathematical Economics 38 (2002) 393–410 Arbitrage and price revelation with asymmetric information and incomplete markets Bernard Cornet a,∗, Lionel De Boisdeffre a,b a CERMSEM, Université de Paris 1, 106–112 boulevard de l’Hˆopital, 75647 Paris Cedex 13, France b Cambridge University, Cambridge, UK Received 5 January 2002; received in revised form 7 September 2002; accepted 10 September 2002 Abstract This paper deals with the issue of arbitrage with differential information and incomplete financial markets, with a focus on information that no-arbitrage asset prices can reveal. Time and uncertainty are represented by two periods and a finite set S of states of nature, one of which will prevail at the second period. Agents may operate limited financial transfers across periods and states via finitely many nominal assets. Each agent i has a private information about which state will prevail at the second period; this information is represented by a subset Si of S. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩iSi, hence assumed to be non-empty. Our analysis is two-fold. We first extend the classical symmetric information analysis to the asym- metric setting, via a concept of no-arbitrage price. Second, we study how such no-arbitrage prices convey information to agents in a decentralized way. The main difference between the symmetric and the asymmetric settings stems from the fact that a classical no-arbitrage asset price (common to every agent) always exists in the first case, but no longer in the asymmetric one, thus allowing arbitrage opportunities. This is the main reason why agents may need to refine their information up to an information structure which precludes arbitrage. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Asymmetric information; Private information; No-arbitrage; Incomplete markets; Refinement; Information revealed by prices 1. Introduction In economies subject to uncertainty and asymmetric information, agents seek to infer relevant information from market indicators, such as prices, to refine their strategies. This ∗ Corresponding author. Tel.: +33-1-55-43-4282; fax: +33-1-55-43-4301. E-mail addresses: [email protected] (B. Cornet), [email protected] (L. De Boisdeffre). 0304-4068/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0304-4068(02)00075-7 394 B. Cornet, L. De Boisdeffre / Journal of Mathematical Economics 38 (2002) 393–410 issue is traditionally tackled by the so-called “rational expectations” models by assuming, quoting Radner (1979), that “agents have a “model” or “expectations” of how equilibrium prices are determined.” In this paper, agents learn from asset prices about partners’ private information, by ana- lyzing the arbitrage opportunities of the financial markets. They need not know the ex ante characteristics of the economy (preferences, endowments of other agents) or a defined rela- tionship between prices and the collection of private information signals in the economy, as in the rational expectations’ models. Thus, they are only required to know the market prices and their own characteristics. We define a notion of equilibrium, embedding the way agents infer information from asset prices, and its properties will be presented in a companion paper. For this purpose, however, we need first to study arbitrage theory with asymmetric information, which is the main aim of the present paper. We consider the simplest tractable setting for the study of arbitrage. Time and uncertainty are represented by two periods (t = 0 and t = 1) and a finite set S of states of nature, one of which will prevail at the second period. Agents may operate limited financial transfers across periods and states via finitely many nominal assets. Each consumer receives a private information signal about which state will prevail at the second period. Asymmetric infor- mation is thus represented, for each agent i, by a subset Si of S. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩iSi, hence assumed to be non-empty. Similarly, when agents refine their information, i.e. when they infer a smaller set Σi ⊂ Si, they also receive no wrong signal, so that ∩iΣi =∅. This is guaranteed, in particular, when the refinement (Σi) of the collection (Si) preserves its pooled information, that is, ∩iSi =∩iΣi. Our analysis is two-fold. We first extend the classical non-arbitrage analysis to the asymmetric setting, via a concept of no-arbitrage price, and second, we study how such no-arbitrage prices convey information to agents in a decentralized way. The main differ- ence between the symmetric and asymmetric settings stems from the fact that a classical no-arbitrage asset price (common to every agent) always exists in the symmetric case, but no longer in the asymmetric one, thus allowing arbitrage opportunities. This is the main reason why agents may need to refine their information up to an information structure precluding arbitrage. The paper is organized as follows. In Section 2, we present the framework and recall the basic concepts of arbitrage-free information structures, refinements, and no-arbitrage prices. We also define the notion of financial equilibrium in an asymmetric setting, which explic- itly presents consumers’ behavior and the need for a refinement of information when it is not arbitrage-free at the outset (Definition 2.2). In Section 3, we characterize arbitrage-free structures by the absence of future (i.e. at t = 1) arbitrage opportunities on the finan- cial market, called the AFAO property (Proposition 3.1). We show that every information ¯ structure (Si) has a unique coarsest arbitrage-free refinement, denoted by (Si), which does ¯ not contain any wrong signals since ∩iSi =∩iSi (Proposition 3.2). We end the section ¯ with the relationship between “fully-revealing” structures, i.e. such that Si coincides with agents’ pooled information, market completeness (Proposition 3.3) and symmetric infor- mation (Proposition 3.4). In Section 4, we first define, for every agent i and every asset price q, the “revealed information set” Si(q) ⊂ Si (Definition 4.1). We then define an extended notion of no-arbitrage asset price (Definition 4.2), as the common no-arbitrage B. Cornet, L. De Boisdeffre / Journal of Mathematical Economics 38 (2002) 393–410 395 prices associated to all arbitrage-free refinements. This allows us to characterize no-arbitrage prices q as those which “reveal” an information structure, i.e. such that ∩iSi(q) =∅ (Definition 4.2 and Proposition 4.2). Finally, we show that the coarsest arbitrage-free refine- ¯ ¯ ment (Si) can always be revealed by some no-arbitrage price q, that is, Si = Si(q) for every i (Proposition 4.3). Some conclusions may be drawn in terms of the financial equilibrium notion presented in the paper (Definition 2.3). When the initial information structure (Si) is arbitrage-free, consumers may keep their initial information sets. Otherwise, they must refine their beliefs up to an arbitrage-free information structure, to be able to perform their maximization ¯ problem (Proposition 2.2). The coarsest arbitrage-free refinement (Si) allows to do it in such a way that it can always be revealed by some price q and it does not contain any wrong ¯ signal since ∩iSi =∩iSi. Hence, agents may always update their beliefs through prices in a decentralized way: neither the presence of another agent, nor the knowledge of the other agents’ characteristics is required.1 2. The model 2.1. The two-period model and financial markets We consider the basic model of a two time-period economy with private information, and nominal assets: the simplest tractable model which allows us to present arbitrage. It is also assumed that there are finite sets I, S, and J, respectively, of agents, states of nature, and nominal assets. In what follows, the first period will also be referred to as t = 0 and the second period, as t = 1. There is an a priori uncertainty at the first period (t = 0) about which of the states of nature s ∈ S will prevail at the second period (t = 1). For the sake of unified notations of time and uncertainty, the non-random state at the first period is denoted by s = 0 and S stands for the set {0}∪S. Similarly, if Σ ⊂ S, Σ will stand for {0}∪Σ. Each agent i ∈ I has a private information at the first period about the possible states of nature of the second period, that is, she knows that the true state will be in a subset Si of S, or, equivalently, that the true state will not belong to the complementary set (in S)ofSi. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩iSi, hence assumed to be non-empty throughout the paper. 1 We shall use hereafter the following notations. If I and J are finite sets, the space RI (identified to R#I whenever necessary) of functions x : I → R (also denoted x = (x(i))i∈I or x = (xi)) is endowed with the √ I Euclidean product x · y := i∈I x(i)y(i), and we denote by x := x · x the Euclidean norm. In R , the notation x ≥ y (respectively x y) means that x(i) ≥ y(i) (respectively x(i) > y(i)) for every i and we let I L I L j R+ ={x ∈ R |x ≥ 0}, R++ ={x ∈ R |x 0}.AnI × J-matrix A = (ai )i∈I,j∈J (identified with a classical I×J j J (#I) × (#J)-matrix if necessary) is an element of R whose rows are denoted A[i] for (ai )j∈J ∈ R (i ∈ I), j j I J I and columns A = (ai )i∈I ∈ R (j ∈ J).